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摘 要:

In this paper we introduce a new notion of collective tree spanners. We say that a graph G =( V, E) admits a system of µ col- lective additive tree r-spanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈T (G) exists such that dT (x, y) ≤ dG(x, y )+ r. Among other results, we show that any chordal graph, chordal bipartite graph or co- comparability graph admits a system of at most log2 n collective addi- tive tree 2-spanners and any c-chordal graph admits a system of at most log2 n collective additive tree (2� c/2� )-spanners. Towards establishing these results, we present a general property for graphs, called (α, r)- decomposition, and show that any (α, r)-decomposable graph G with n vertices admits a system of at most log1/α n collective additive tree 2r- spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs.